The ML-EM algorithm in continuum: sparse measure solutions

TL;DR

This paper analyzes the ML-EM algorithm for linear inverse problems with Poisson noise, revealing conditions under which solutions are sparse or smooth, and explaining the spiky images often observed in practice.

Contribution

It provides a measure-theoretic analysis of ML-EM, showing sparsity in short exposure regimes and smoothness in long exposure regimes, with concentration bounds.

Findings

Sparse solutions occur when measurements are outside the cone of feasible images.

Long exposure leads to measures without singular parts.

Concentration bounds quantify the probability of sparse solutions.

Abstract

Linear inverse problems $A \mu = \delta$ with Poisson noise and non-negative unknown $\mu \geq 0$ are ubiquitous in applications, for instance in Positron Emission Tomography (PET) in medical imaging. The associated maximum likelihood problem is routinely solved using an expectation-maximisation algorithm (ML-EM). This typically results in images which look spiky, even with early stopping. We give an explanation for this phenomenon. We first regard the image $\mu$ as a measure. We prove that if the measurements $\delta$ are not in the cone $\{A \mu, \mu \geq 0\}$, which is typical of short exposure times, likelihood maximisers as well as ML-EM cluster points must be sparse, i.e., typically a sum of point masses. On the other hand, in the long exposure regime, we prove that cluster points of ML-EM will be measures without singular part. Finally, we provide concentration bounds for the probability to be in the sparse case.